a current of 40 microampere is passed through a solution of silver nitrate for 32 minute using Pt electrodes.an uniform single atom thick layer of Agis deposited covering 43%cathode surface.what is the total surface area of cathode if each Ag atom covers 5.4 into 10 to the power _16 square cm.

To determine the total surface area of the cathode based on the given information, we need to follow a series of logical steps that involve some calculations related to the deposition of silver (Ag) from silver nitrate (AgNO3) solution. Let's break this down step by step.

Understanding the Problem

We know the following:

• Current (I) = 40 microamperes = 40 x 10^-6 A
• Time (t) = 32 minutes = 32 x 60 seconds = 1920 seconds
• Percentage of cathode surface covered by Ag = 43%
• Area covered by each Ag atom = 5.4 x 10^-16 cm²

Calculating the Total Charge

First, we need to calculate the total charge (Q) that has passed through the solution during the deposition process. The formula for charge is:

Q = I × t

Substituting the values:

Q = (40 x 10^-6 A) × (1920 s) = 76.8 x 10^-3 C = 0.0768 C

Determining Moles of Silver Deposited

Next, we can find out how many moles of silver have been deposited using Faraday's law of electrolysis. The molar mass of silver (Ag) is approximately 107.87 g/mol, and the charge required to deposit one mole of silver is 96500 C (Faraday's constant).

Using the formula:

n = Q / F

where n is the number of moles, Q is the total charge, and F is Faraday's constant:

n = 0.0768 C / 96500 C/mol ≈ 7.95 x 10^-7 mol

Calculating the Number of Silver Atoms

Now, we can find the number of silver atoms deposited. Using Avogadro's number (approximately 6.022 x 10²³ atoms/mol):

Number of atoms = n × Avogadro's number

Number of atoms = (7.95 x 10^-7 mol) × (6.022 x 10²³ atoms/mol) ≈ 4.78 x 10¹⁷ atoms

Calculating the Area Covered by Silver Atoms

Next, we calculate the total area covered by these silver atoms:

Total area covered = Number of atoms × Area per atom

Total area covered = (4.78 x 10¹⁷ atoms) × (5.4 x 10^-16 cm²) ≈ 2.58 x 10² cm²

Finding the Total Surface Area of the Cathode

Since this area represents only 43% of the total cathode surface area, we can find the total surface area (A_total) using the formula:

A_total = Area covered / Percentage covered

A_total = (2.58 x 10² cm²) / 0.43 ≈ 6.00 x 10² cm²

Final Result

Thus, the total surface area of the cathode is approximately 600 cm². This calculation illustrates how electrochemical principles can be applied to determine physical properties of materials based on their behavior in solution.